Abstract-We introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses a discrete weighted p-Dirichlet energy, and an approximation one. This is the discrete analogue of recent continuous Euclidean nonlocal regularization functionals. The proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted p-Laplace operator, parameterized by the degree p of regularity, the graph structure and the graph weight function. These discrete processing methods provide a graph-based version of recently proposed semi-local or nonlocal processing methods used in image and mesh processing, such as the bilateral filter, the TV digital filter or the nonlocal means filter. It works with equal ease on regular 2D-3D images, manifolds or any data. We illustrate the abilities of the approach by applying it to various types of images, meshes, manifolds and data represented as graphs.

International audienceIn this paper we introduce a new family of partial difference operators on graphs and study equations involving these operators. This family covers local variational $p$-Laplacian, $\infty$-Laplacian, nonlocal $p$-Laplacian and $\infty$-Laplacian, $p$-Laplacian with gradient terms, and gradient operators used in morphology based on the partial differential equation. We analyze a corresponding parabolic equation involving these operators which enables us to interpolate adaptively between $p$-Laplacian diffusion-based filtering and morphological filtering, i.e., erosion and dilation. Then, we consider the elliptic partial difference equation with its corresponding Dirichlet problem and we prove the existence and uniqueness of respective solutions. For $p=\infty$, we investigate the connection with Tug-of-War games. Finally, we demonstrate the adaptability of this new formulation for different tasks in image and point cloud processing, such as filtering, segmentation, clustering, and inpainting

In this paper we propose an adaptation of the Eikonal equation on weighted graphs, using the framework of Partial difference Equations, and with the motivation of extending this equation's applications to any discrete data that can be represented by graphs. This adaptation leads to a finite difference equation defined on weighted graphs and a new efficient algorithm for multiple labels simultaneous propagation on graphs, based on such equation. We will show that such approach enables the resolution of many applications in image and high dimensional data processing using a unique framework. Keywords Eikonal equation • weighted graph • non-local image processing • active contour • PdE • fast marching • high dimensional data This work was supported under a doctoral grant of the Conseil Régional de Basse-Normandie and the Coeur et Cancer association.